Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/31)

The rewrite relation of the following TRS is considered.

app(app(:,app(app(:,x),y)),z)	→	app(app(:,x),app(app(:,y),z))	(1)
app(app(:,app(app(+,x),y)),z)	→	app(app(+,app(app(:,x),z)),app(app(:,y),z))	(2)
app(app(:,z),app(app(+,x),app(f,y)))	→	app(app(:,app(app(g,z),y)),app(app(+,x),a))	(3)
app(app(map,f),nil)	→	nil	(4)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(5)
app(app(filter,f),nil)	→	nil	(6)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(f,x)),f),x),xs)	(7)
app(app(app(app(filter2,true),f),x),xs)	→	app(app(cons,x),app(app(filter,f),xs))	(8)
app(app(app(app(filter2,false),f),x),xs)	→	app(app(filter,f),xs)	(9)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

app^#(app(:,app(app(:,x),y)),z)	→	app^#(app(:,x),app(app(:,y),z))	(10)
app^#(app(:,app(app(:,x),y)),z)	→	app^#(app(:,y),z)	(11)
app^#(app(:,app(app(:,x),y)),z)	→	app^#(:,y)	(12)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(app(+,app(app(:,x),z)),app(app(:,y),z))	(13)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(+,app(app(:,x),z))	(14)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(app(:,x),z)	(15)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(:,x)	(16)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(app(:,y),z)	(17)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(:,y)	(18)
app^#(app(:,z),app(app(+,x),app(f,y)))	→	app^#(app(:,app(app(g,z),y)),app(app(+,x),a))	(19)
app^#(app(:,z),app(app(+,x),app(f,y)))	→	app^#(:,app(app(g,z),y))	(20)
app^#(app(:,z),app(app(+,x),app(f,y)))	→	app^#(app(g,z),y)	(21)
app^#(app(:,z),app(app(+,x),app(f,y)))	→	app^#(g,z)	(22)
app^#(app(:,z),app(app(+,x),app(f,y)))	→	app^#(app(+,x),a)	(23)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(24)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(25)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(26)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(27)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(28)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(f,x)),f),x)	(29)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter2,app(f,x)),f)	(30)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(filter2,app(f,x))	(31)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(32)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(cons,x),app(app(filter,f),xs))	(33)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(cons,x)	(34)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(35)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(filter,f)	(36)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(37)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(filter,f)	(38)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(27)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(26)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(32)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(35)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(37)

1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[app(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[map]	=	0
[cons]	=	0
[filter]	=	0
[filter2]	=	0
[true]	=	0
[false]	=	0
[app^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(32)

1	>	1
2	>	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(26)

1	>	1
2	>	2
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(27)

1	≥	1
2	>	2
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(35)

2	≥	2
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(37)

2	≥	2

As there is no critical graph in the transitive closure, there are no infinite chains.

The 2^nd component contains the pair

app^#(app(:,app(app(:,x),y)),z)	→	app^#(app(:,y),z)	(11)
app^#(app(:,app(app(:,x),y)),z)	→	app^#(app(:,x),app(app(:,y),z))	(10)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(app(:,x),z)	(15)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(app(:,y),z)	(17)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

prec(app)	=	2	weight(app)	=	1
prec(:)	=	0	weight(:)	=	1
prec(+)	=	1	weight(+)	=	2

in combination with the following argument filter

π(app^#)	=	1
π(app)	=	[1,2]
π(:)	=	[]
π(+)	=	[]

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

app^#(app(:,app(app(:,x),y)),z)	→	app^#(app(:,y),z)	(11)
app^#(app(:,app(app(:,x),y)),z)	→	app^#(app(:,x),app(app(:,y),z))	(10)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(app(:,x),z)	(15)
app^#(app(:,app(app(+,x),y)),z)	→	app^#(app(:,y),z)	(17)

could be deleted.

1.1.2.1 P is empty

There are no pairs anymore.